On the Bombieri–Korobov estimate for Weyl sums

Parsell, Scott T.

doi:10.4064/aa138-4-7

Cited by 9 publications

(15 citation statements)

References 2 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…On the other hand, we note that using bounds of exponential sums obtained with the method of Vinogradov instead of Lemma 11, see [5,15,30,37] and references therein, also leads to some nontrivial on J f (M; R, S) but these results seem to be weaker than a combination of Theorem 5 with the bounds from [12].…”

Section: Commentsmentioning

confidence: 91%

See 1 more Smart Citation

Points on curves in small boxes and applications

Chang¹,

Cilleruelo²,

Garaev³

et al. 2014

Michigan Math. J.

View full text Add to dashboard Cite

Abstract. We introduce several new methods to obtain upper bounds on the number of solutions of the congruenceswith a prime p and a polynomial f , where (x, y) belongs to an arbitrary square with side length M . We use these results and methods to derive non-trivial upper bounds for the number of hyperelliptic curvesover the finite field F p of p elements, with coefficients in a 2g-dimensional cubethat are isomorphic to a given curve and give an almost sharp lower bound on the number of non-isomorphic hyperelliptic curves with coefficients in that cube. Furthermore, we study the size of the smallest box that contain a partial trajectory of a polynomial dynamical system over F p .

show abstract

Section: Commentsmentioning

confidence: 91%

“…By (29), (30), (31), the absolute value of the both hand side is bounded by pM 1+o(1) T −1 . Thus, we get…”

Section: 1mentioning

confidence: 99%

Points on curves in small boxes and applications

Chang¹,

Cilleruelo²,

Garaev³

et al. 2014

Michigan Math. J.

View full text Add to dashboard Cite

show abstract

“…For large values of k, meanwhile, one had the work of Ford [6]. Together with refinements for intermediate values of k due to Parsell [9] and Boklan and Wooley [4], this delivered the bounds G(9) 365, . .…”

Section: Introductionmentioning

confidence: 99%

Vinogradov’s mean value theorem via efficient congruencing, II

Wooley¹

2013

Duke Math. J.

View full text Add to dashboard Cite

We apply the efficient congruencing method to estimate Vinogradov's integral for moments of order 2s, with 1 s k 2 − 1. Thereby, we show that quasi-diagonal behaviour holds when s = o(k 2 ), we obtain near-optimal estimates for 1 s 1 4 k 2 + k, and optimal estimates for s k 2 − 1. In this way we come half way to proving the main conjecture in two different directions. There are consequences for estimates of Weyl type, and in several allied applications. Thus, for example, the anticipated asymptotic formula in Waring's problem is established for sums of s kth powers of natural numbers whenever s 2k 2 − 2k − 8 (k 6).

show abstract

“…Taking β = 2 − δ k in Proposition 1 and u = 2k and η = 1 − δ k /2 in Lemma 2 proves the following theorem. size, recent records have been set by Ford [4], recently improved by Parsell [18], and further sharpened by Boklan and Wooley [2]. For sufficiently large k, the current best asymptotic result, due to Ford [4], states thatG(k) k 2 (log k + log log k + O(1)).…”

Section: 2mentioning

confidence: 99%

On discrete fractional integral operators and mean values of Weyl sums

Pierce¹

2011

Bulletin of the London Mathematical Society

View full text Add to dashboard Cite

In this paper, we prove new p → q bounds for a discrete fractional integral operator by applying techniques motivated by the circle method of Hardy and Littlewood to the Fourier multiplier of the operator. From a different perspective, we describe explicit interactions between the Fourier multiplier and mean values of Weyl sums. These mean values express the average behaviour of the number r s,k (l) of representations of a positive integer l as a sum of s positive kth powers. Recent deep results within the context of Waring's problem and Weyl sums enable us to prove a further range of complementary results for the discrete operator under consideration. The weak-type L r space is the space of functions such that |{θ : |f (θ)| > α}| cα −r for all α > 0, which when normed becomes the Lorentz space L r,∞ .

show abstract

On the Bombieri–Korobov estimate for Weyl sums

Cited by 9 publications

References 2 publications

Points on curves in small boxes and applications

Points on curves in small boxes and applications

Vinogradov’s mean value theorem via efficient congruencing, II

On discrete fractional integral operators and mean values of Weyl sums

Contact Info

Product

Resources

About